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We study the problem of learning equivariant neural networks via gradient descent. The incorporation of known symmetries ("equivariance") into neural nets has empirically improved the performance of learning pipelines, in domains ranging from biology to computer vision. However, a rich yet separate line of learning theoretic research has demonstrated that actually learning shallow, fully-connected (i.e. non-symmetric) networks has exponential complexity in the correlational statistical query (CSQ) model, a framework encompassing gradient descent. In this work, we ask: are known problem symmetries sufficient to alleviate the fundamental hardness of learning neural nets with gradient descent? We answer this question in the negative. In particular, we give lower bounds for shallow graph neural networks, convolutional networks, invariant polynomials, and frame-averaged networks for permutation subgroups, which all scale either superpolynomially or exponentially in the relevant input dimension. Therefore, in spite of the significant inductive bias imparted via symmetry, actually learning the complete classes of functions represented by equivariant neural networks via gradient descent remains hard. 

Abstract We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank–Wolfe (Fw) methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: the computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.